Fully inert submodules of torsion-free modules over the ring of p-adic integers

B. Goldsmith; L. Salce; P. Zanardo

Displaying similar documents to “Fully inert submodules of torsion-free modules over the ring of p-adic integers”

An addendum and corrigendum to "Almost free splitters" (Colloq. Math. 81 (1999), 193-221)

Rüdiger Göbel, Saharon Shelah (2001)

Colloquium Mathematicae

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Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if $E x t ¹_{R} (G, G) = 0$ . In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

Cellular covers of cotorsion-free modules

Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann (2012)

Fundamenta Mathematicae

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In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $π ⁎ : H o m_{R} (G, G) ≅ H o m_{R} (G, H)$ , where π⁎(φ) = πφ for each $φ \in H o m_{R} (G, G)$ (where maps are acting on the left). On...

T-Rickart modules

S. Ebrahimi Atani, M. Khoramdel, S. Dolati Pish Hesari (2012)

Colloquium Mathematicae

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We introduce the notions of T-Rickart and strongly T-Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that R is right Σ-t-extending if and only if every R-module is T-Rickart. Also, every free R-module is T-Rickart if and only if $R = Z ₂ (R_{R}) \oplus R^{'}$ , where R’ is a hereditary right R-module. Examples illustrating the results are presented.

Precovers and Goldie’s torsion theory

Ladislav Bican (2003)

Mathematica Bohemica

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Recently, Rim and Teply , using the notion of $τ$ -exact modules, found a necessary condition for the existence of $τ$ -torsionfree covers with respect to a given hereditary torsion theory $τ$ for the category $R$ -mod of all unitary left $R$ -modules over an associative ring $R$ with identity. Some relations between $τ$ -torsionfree and $τ$ -exact covers have been investigated in . The purpose of this note is to show that if $σ = (𝒯_{σ}, ℱ_{σ})$ is Goldie’s torsion theory and $ℱ_{σ}$ is a precover class, then $ℱ_{τ}$ is a precover class...

A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

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There is a classical result known as Baer’s Lemma that states that an $R$ -module $E$ is injective if it is injective for $R$ . This means that if a map from a submodule of $R$ , that is, from a left ideal $L$ of $R$ to $E$ can always be extended to $R$ , then a map to $E$ from a submodule $A$ of any $R$ -module $B$ can be extended to $B$ ; in other words, $E$ is injective. In this paper, we generalize this result to the category $q_{ω}$ consisting of the representations of an infinite line quiver. This generalization of Baer’s...

Precobalanced and cobalanced sequences of modules over domains

Anthony Giovannitti, H. Pat Goeters (2007)

Mathematica Bohemica

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The class of pure submodules ( $𝒫$ ) and torsion-free images ( $ℛ$ ) of finite direct sums of submodules of the quotient field of an integral domain were first investigated by M. C. R. Butler for the ring of integers (1965). In this case $𝒫 = ℛ$ and short exact sequences of such modules are both prebalanced and precobalanced. This does not hold for integral domains in general. In this paper the notion of precobalanced sequences of modules is further investigated. It is shown that as in the case for...

$\oplus$ -cofinitely supplemented modules

H. Çalışıcı, A. Pancar (2004)

Czechoslovak Mathematical Journal

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Let $R$ be a ring and $M$ a right $R$ -module. $M$ is called $\oplus$ -cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$ . In this paper various properties of the $\oplus$ -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$ -cofinitely supplemented modules is $\oplus$ -cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$ -module is $\oplus$ -cofinitely supplemented. In addition, if $M$ has the...

$τ$ -supplemented modules and $τ$ -weakly supplemented modules

Muhammet Tamer Koşan (2007)

Archivum Mathematicum

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Given a hereditary torsion theory $τ = (𝕋, 𝔽)$ in Mod- $R$ , a module $M$ is called $τ$ -supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A / C$ $τ -$ torsion. A submodule $V$ of $M$ is called $τ$ -supplement of $U$ in $M$ if $U + V = M$ and $U \cap V \leq τ (V)$ and $M$ is $τ$ -weakly supplemented if every submodule of $M$ has a $τ$ -supplement in $M$ . Let $M$ be a $τ$ -weakly supplemented module. Then $M$ has a decomposition $M = M_{1} \oplus M_{2}$ where $M_{1}$ is a semisimple module and $M_{2}$ is a module with $τ (M_{2}) \leq_{e} M_{2}$ . Also, it is shown that; any finite sum of $τ$ -weakly...